Magic Stars Index Page

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I hope you find find this page interesting and informative. I will be adding to it from my notes and future studies as time permits so please come back often.

So far, I have been concentrating mainly on finding the basic solutions for the different orders. There is much left to discover about the characteristics of the individual orders. Share with me the excitement of the search.

If you are also interested in Magic Stars, I would like to hear from you.

March 1, 2005: NEW 4 pages added. See bottom of contents!
April 4, 2007: NEW(page 9x5 stars) added.
May 18,2007: New section on early solvers added to Order 6 Stars.
2010 Note: Because this section was a prominent part of the original web site, I have chosen to include it on this renamed Magic Hypercubes site.

On this page

Introduction A basic definition of Magic Stars and the similarity to Magic Squares. Includes a diagram of the 3 Order-9 patterns and shows the order that numbers are assigned to the lines.
Basic Equivalent Solutions An explanation of which solutions are considered basic and which are equivalent solutions.
The two requirements for a basic solution and converting an equivalent to a basic solution.
Complements Index Numbers Each solution has a complement. If the solution is basic the complement is an equivalent and must be normalized to arrive at it's basic solution.
Includes a diagram of four Order-6 solutions to illustrate the above.
References for Magic Stars Sixteen different diagrams from Order-5 to Order-11d
Also shown is the solution number and the total number of solutions.

Other pages in this section

1. A Magic Star Definition What is a Magic Star? Here is a formal definition and an explanation of terms used in my discussion of magic stars.
Included also are comparisons between the different orders.
2. Examples of Magic Stars Sixteen different diagrams from Order-5 to Order-11d
Also shown is the solution number and the total number of solutions.
3. Examples of Magic Stars - 2 Sixteen different diagrams from Order-12a to Order-14e.
Also shown is the solution number and the estimated total number of solutions.
4. Big Magic Stars 1 solution for pattern A of orders 15 to 20. Also blank graphs of the other patterns for each order.
5. Order-5 Magic Stars Order-5 is not a pure magic star but there are 12 solutions using numbers 1 to 12 but omitting numbers 7 and 11. Another 12 solutions leave out the 2 and 6.
6. Order-6 Magic Stars A list of the 80 basic solutions along with characteristics. 20 sets of 4. Super-magic stars. A tribute to H. E. Dudeney.
7. Order-6 Solution List A tabular listing of the 80 basic solutions along with additional features.
8. H. E. Dudeney Features Features of the order-6 magic stars that were first reported by H. E. Dudeney in 1926.
9. Order-7 Magic Stars General characteristics. Lists of the 72 basic solutions for each of the 2 patterns.
10. Order-8 Magic Stars General characteristics. Lists of the 112 basic solutions for each of the 2 patterns.
11. Order-9 Magic Stars General characteristics. Condensed lists of basic solutions for each of the 3 patterns.
12. Order-10 Magic Stars General characteristics. Condensed lists of basic solutions for each of the 3 patterns.
13. Order-11 Magic Stars General characteristics. Condensed lists of basic solutions for each of the 4 patterns.
14. Prime Magic Stars Magic Stars consisting of prime numbers. Lists of minimal solutions consecutive primes solutions for orders 5 and 6.
15. Prime Magic Stars - 2 Diagrams and lists of minimal solutions consecutive primes solutions for orders 7 A B and 8 A B.
16. Unusual Magic Stars Patterns with combinations of stars or more then 4 numbers per line.
17. Iso-like Magic Stars Stars that are transformations of magic squares. Also plusmagic and diammagic squares.
18. Magic 9 x 5 Order-6 Stars This new page (April 2007) shows 1 solution for each magic constant (S) of 46 to 54 of the 5 numbers in each of 9 lines magic hexagram. Note that most magic star pages on this site deal with only 4 numbers/line.
19. 3-D Magic Stars This magic 8-point star contains 12 lines of 3 numbers, plus many other lines as a result of the missing numbers of the series forming a nucleus and two satellites.
20. Trenkler Stars Marian Trenkler defines stars as of 2 types. He also defines almost-magic weakly-magic stars.
21. Magic Star Puzzles Pictures showing star (and other magic object) puzzles. Some quite old.
Also, some pencil-and-paper puzzles of magic objects.
22. Star Updates This page, started in March, 2005, will contain material added to this site or links to sub-pages of such material.
a. Simon WhiteChapel With emails starting in 2001, Simon presents solutions for pattern A of magic stars from 15 to 100.
b. Jon Wharf With emails starting in 2003, Jon confirms the total solution count for all orders and patterns from 6 to11, and provides the total solution count all order 12 patterns. He also supplies some solutions for all patterns of orders 13 and 14.
c. Andrew Howroyd First contacted me in February, 2005. He also confirms all total solution counts, and investigated permutations between patterns of orders 10 and 11.

Introduction

Magic stars are similar to Magic Squares in many ways. The order refers to the number of points in the pattern. A standard magic star always contains 4 numbers in each line and in a pure magic star they consist of the series from 1 to 2n where n is the order of the star.

The diagram above demonstrates also how the numbers are assigned to the cells one line at a time.
Note also, all orders greater then six consist of multiple patterns, each of which consist of a different list of basic solutions. I have found no reference in the literature to this fact

Of course, some star patterns have more then two line crossings (plus the two points) per line. See, for example, orders 9b and 9c above. In these cases, we could assign more then 4 numbers to a line in such a way that all lines sum the same. These too would be magic stars. However, to keep the variations to a manageable number, my studies have been limited to the cases where only the perimeter line junctions (i.e. the points and valleys) have numbers assigned to them.

Pattern naming convention. Originally I had rather arbitrarily assigned names a, b, c, etc to the various patterns of an order of magic star. In January, 2001, Aale de Winkel suggested a systematical way of applying these labels.
Imagine the points of a star diagram as being points on a circle. Then each point in turn is connected by a line to another point, by moving around the circle clockwise. If we step once and connect to the second point, the pattern is called 'A'. Stepping twice, and connecting to the third point, produces pattern 'B'. etc.
Another way to look at this subject:
'A' has 4 intersections per line, 'B' has 6, 'C' has 8, 'D' has 10, and 'E' (required for orders 13 and 14) has 12 intersections per line.

By Feb. 16, 2001, all relevant pages have been revised to show the new pattern names.

Basic Equivalent Solutions

Each star has solutions that are apparently different but in fact are only rotations and/or reflections of the basic solution. The order-10 star with its 10 degrees of rotational symmetry, each of which may be reflected, has 20 apparently different solutions. Only one of these is considered the basic solution.

Two characteristics determine the Basic Solution.

  • The top point of the diagram has the lowest value of all the points.

  • The valley to the right of the top point has a lower value then that of the valley to the left.

Any magic star solution may be converted to a basic solution by normalizing it, i.e. performing the necessary rotations and/or reflections so the solution confirms to the above criteria.

Any magic star can be converted to another magic star by adding or multiplying each number in the star by a constant. This feature also applies to magic squares.
Of course, the resulting star would not be pure (normal) because the number series would no longer be consecutive.

Complements Index Numbers

Any magic star can be made into another magic star by complementing each number of the original star in turn. This is done by subtracting each number from n + 1. In the case of the order-6 star, which uses the numbers 1 to 12, you subtract each number from 13 to obtain the new number.

Diag-ram Solut-ion # a b c d e f g h i j k l Compl.
Sol. #
Compl.
Pair #
description
A. 38 1 9 11 5 4 10 7 6 12 3 8 2 79 32 How solutions are written
B. 39 1 9 12 4 3 11 8 7 10 5 6 2 78 33 The next solution in index order
C. 12 4 1 9 10 2 5 6 3 8 7 11 Not a basic solution
D. 78 5 2 10 9 1 4 12 3 6 7 8 11 39 33 Diagram c. normalized by rotation 2 positions clockwise, then a horizontal reflection

If the original is a basic solution, the complement star will not be a basic solution. It is an equivalent, but after normalizing, it will be another basic solution. When enumerating solutions for magic squares, the complements are also counted as basic solutions. We will follow the same convention when counting and indexing the magic star solutions. This means that the number of solutions for each order of magic star must always be an even number and the number of complement pairs is exactly half the number of total solutions. To put it another way, all basic solutions come in pairs which are complements of each other.

The fact that all solutions have a pair partner determine some characteristics for a particular order. For example, if you find a solution with all odd numbers at the points, you can be confident another solution exists that has all even numbers at the points. Likewise, if a solution exists that has all the low numbers at the points, another one exists that has all the high numbers.

The complementing process works for all magic squares and all magic stars even if the numbers are not consecutive or do not start at 1. In such cases, the complementary number is obtained by subtracting from the sum of the first and last number in the series used. Even prime magic stars have a compliment, although because compliments of many of the prime numbers are not prime numbers, the resulting magic star will not be a prime magic star.

Order-5 magic stars come in pairs where the points of one member appear as the valleys of the other member. I call these pairs Pcomp because they are complements of each other, but not in the accepted sense.

References for Magic Stars

Order-6 is the smallest pure magic star and the only one with only one star pattern (a fact not mentioned in the literature). In fact, in contrast to the voluminous literature for magic squares spanning 100's of years, there has been very little published on magic stars. The two main sources of information I have been able to locate are:

  • H.E.Dudeney, 536 Puzzles Curious Problems, Scribner's 1967. Lots of info on order-6.

  • Martin Gardner, Mathematical Recreations column of Scientific American, Dec. 1965, reprinted with addendum in Martin Gardner, Mathematical Carnival, Alfred A. Knoff, 1975. Mostly on order 6, but mention made of total basic solutions for orders 7 8 (also corrected number for order-6).

Marián Trenkler of Safarik University, Kosice, Slovakia published a paper on Magic Stars.
It is called Magicke hviezdy (Magic stars) and appeared in Obsory matematiky, fyziky a informatiky, 51(1998), pages 1-7. (Obsory = horizons (or line of sight) of mathematics, physics and informatics.

Magic squares, perhaps because they are quite ordered structures, have been studied for centuries. In contrast, magic stars have few similarities between orders, or for that matter even between patterns within an order. This makes it necessary to study each pattern individually

My studies (to March 1998) include all basic solutions for orders 5 to 11 and most solutions for order-12, a total of 20 patterns.
Also, many solutions for each of the 10 patterns of orders 13 and 14.

Here are 16 sample magic stars for all orders and patterns from five to eleven and here are the 14 patterns for orders twelve to fourteen. And here are 6 examples of pattern A stars of orders 15 to 20.
Also, be sure to check out Definitions and Details, and the Order-6 page. Over time, I intend to add more pages, covering details of the different orders, and including lists of solutions. So please check this site periodically.

This page was originally posted March 1998
It was last updated July 23, 2011
Harvey Heinz   harveyheinz@shaw.ca
Copyright © 1998-2009 by Harvey D. Heinz